Coupling local and nonlocal evolution equations

Abstract

We prove existence, uniqueness and several qualitative properties for evolution equations that combine local and nonlocal diffusion operators acting in different subdomains and coupled in such a way that the resulting evolution equation is the gradient flow of an energy functional. We deal with the Cauchy, Neumann and Dirichlet problems, in the last two cases with zero boundary data. For the first two problems we prove that the model preserves the total mass. We also study the decay rates of the solutions for large times. Finally, we show that we can recover the usual heat equation (local diffusion) in a limit procedure when we rescale the nonlocal kernel in a suitable way.

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Acknowledgements

The first two authors were partially supported by the Spanish project MTM2017-87596-P, and the third one by CONICET grant PIP GI No 11220150100036CO (Argentina), by UBACyT grant 20020160100155BA (Argentina) and by the Spanish project MTM2015-70227-P. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant Agreement No. 777822. Part of this work was done during visits of JDR to Madrid and of AG and FQ to Buenos Aires. The authors want to thank these institutions for the nice and stimulating working atmosphere.

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Correspondence to Fernando Quirós.

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Gárriz, A., Quirós, F. & Rossi, J.D. Coupling local and nonlocal evolution equations. Calc. Var. 59, 112 (2020). https://doi.org/10.1007/s00526-020-01771-z

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Mathematics Subject Classification

  • 35K55
  • 35B40
  • 35A05